Confused with the definition of hitting time (Markov chains)

Let $$\emptyset \neq A \subseteq \mathcal{S}$$ on a state space $$\mathcal{S}$$ of a Markov chain $$\{X_n\}_{n=1}^\infty$$. We define

$$T_A := \inf\{n \geq 1 \mid X_n \in A\}$$

with $$\inf \emptyset = +\infty$$?

How do I interpret this infinum? In particular, what does the $$X_n \in A$$ mean?

Is it short written for

$$T_A = \inf\{n \geq 1 \mid X_n(\Omega) \cap A \neq\emptyset\}$$ if we work on the probability space $$(\Omega, \mathcal{F}, \mathbb{P})$$?

Is $$T_A$$ a random variable? I see that one calculates probabilities like $$\mathbb{P}(T_y < \infty)$$ and expectations $$\mathbb{E}T_y$$ so I think it should be a random variable.

Or maybe, it is

$$T_A(\omega) := \inf \{n \geq 1\mid X_n(\omega) \in A\}$$ for $$\omega \in \Omega$$? Can someone clarify?

$$T_A$$ is a random variable (more precisely a stopping time) defined by $$\forall\omega\in\Omega,\quad T_A(\omega)=\inf\{n\ge1\mid X_n(\omega)\in A\}.$$